Let $y=y(x)$ be the solution of the differential equation $x \sin \left(\frac{y}{x}\right) d y=\left(y \sin \left(\frac{y}{x}\right)-x\right) d x, y(1)=\frac{\pi}{2}$ and let $\alpha=\cos \left(\frac{y\left(e^{12}\right)}{e^{12}}\right)$. Then the number of integral value of $p$, for which the equation $x^2+y^2-2 p x+2 p y+\alpha+2=0$ represents a circle of radius $r \leq 6$, is $\_\_\_\_$ .